GMAT Math : Simplifying Algebraic Expressions

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Example Question #1 : Simplifying Algebraic Expressions

Factor $\frac{x^{2}+6x+5}{x^{2}+10x+25}.$

Possible Answers:

$\frac{x+1}{x-5}$

$\frac{1}{(x-5)^{2}}$

$\frac{1}{(x+5)^{2}}$

$\dpi{100} \small 1$

$\frac{x+1}{x+5}$

Correct answer:

$\frac{x+1}{x+5}$

Explanation:

Let's first look at the numerator and denominator separately.

$x^{2}+6x+5$ : We need two numbers that multiply to 5 and add to 6. The numbers 1 and 5 work. So, $x^{2}+6x+5 = (x+5)(x+1)$

$x^{2}+10x + 25$ : We need two numbers that multiply to 25 and add to 10. The numbers 5 and 5 work. So, $x^{2}+10x + 25 = (x+5)(x+5)$

Putting this together, $\frac{x^{2}+6x+5}{x^{2}+10x+25} = \frac{(x+5)(x+1)}{(x+5)(x+5)} = \frac{x+1}{x+5}$

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Example Question #1 : Simplifying Algebraic Expressions

Find the solutions to the equation x+3y+x-3=2y+2x+y .

Possible Answers:

no solution

x=3y

x=2,y=0

x=3, y is any real number

all real numbers

Correct answer:

no solution

Explanation:

Let's combine like terms.

2x+3y-3=2x+3y

-3=0 , so the equation has no solution.

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Example Question #1 : Simplifying Algebraic Expressions

$P = \frac{27D}{2d - D}$

Solve for .

Possible Answers:

2dP + 27

$\frac{27}{2}$

$\frac{2dP}{27+P}$

$\frac{27+P}{2dP}$

2d + 27P

Correct answer:

$\frac{2dP}{27+P}$

Explanation:

You have to isolate by moving around the separate components in the problem. The steps should go as follows:

$P = \frac{27D}{2d - D}$

(2d - D)* P = 27D

2dP - DP = 27D

2dP = 27D + DP

2dP = D*(27 + P)

$\frac{2dP}{27 + P} = D$

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Example Question #4 : Simplifying Algebraic Expressions

Let and be unknown variables. Simplify the following expression: (12X - 2X- 22Y - 32XY) - (42X+62Y-52XY)

Possible Answers:

-32X-84Y-84XY

-32X - 84 Y +20 XY

-32X+40Y-84XY

-32X+40Y-20XY

52X+40Y+20XY

Correct answer:

-32X - 84 Y +20 XY

Explanation:

To simplify algebraically, we combine like terms. First, we should get the expression in one long string, by removing the parentheses. So remembering the communitive property, the first group in parentheses will have no changes when we remove the parentheses. So (12X - 2X- 22Y - 32XY) - (42X+62Y-52XY) simplifies to 12X - 2X- 22Y - 32XY - (42X+62Y-52XY)

However, note the second group in parentheses is being subtracted. So we must invert all the signs in the group to simplify properly. So the previous expression simplifies to

12X - 2X- 22Y - 32XY - 42X-62Y+52XY

Finally we reorder and combine like terms to get

12X - 2X- 42X- 22Y -62Y- 32XY +52XY = -32X-84Y + 20XY

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Example Question #1 : Simplifying Algebraic Expressions

A number is divided by 4; its decimal point is then moved to the right 3 places. This is the same as doing what to the number?

Possible Answers:

Dividing it by 250.

Multiplying it by 2,500.

Dividing it by 4,000.

Dividing it by 400.

Multiplying it by 250.

Correct answer:

Multiplying it by 250.

Explanation:

The best way to illustrate the answer to this question is to do these operations to the number 1.

First, divide by 4:

$1 \div 4 = 0.250$

Now move the decimal point right three spaces:

$0.250\Rightarrow 250.$

This has the effect of multiplying the number by 250.

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Example Question #6 : Simplifying Algebraic Expressions

Which of these expressions is equal to $\log_{7} x^{4}$ ?

Possible Answers:

$\frac{7 \ln x }{4}$

$4 \ln \frac{x}{7}$

$\frac{4 \ln x}{\ln 7}$

$\frac{4 \ln x}{7}$

$\frac{ \ln x +\ln4}{\ln 7}$

Correct answer:

$\frac{4 \ln x}{\ln 7}$

Explanation:

$\log_{7} x^{4} = 4 \log_{7} x = 4 \cdot\frac{ \ln x }{\ln 7} =\frac{ 4\ln x }{\ln 7}$

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Example Question #7 : Simplifying Algebraic Expressions

The sum of three consecutive integers is 12. What is the value of the middle integer?

Possible Answers:

Correct answer:

Explanation:

Let the value of the first integer be . This means that the consecutive integers will be , N+1 , and N+2 . The sum must be 12 which means that

$N+(N+1)+(N+2)=12\Rightarrow 3N+3=12\Rightarrow N=3$

Since N=3 the consecutive integers are 3, 4, and 5. The middle integer is 4.

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Example Question #231 : Algebra

Solve for .

$y=\frac{3x}{2+x}$

Possible Answers:

$x=\frac{-2y}{y-3}$

$x=\frac{2y}{y+3}$

$x=\frac{-2y}{y+3}$

$x=\frac{2y}{y-3}$

Correct answer:

$x=\frac{-2y}{y-3}$

Explanation:

$y=\frac{3x}{2+x}\Rightarrow y(2+x)=3x$

$2y+xy=3x\Rightarrow xy-3x=-2y$

$x(y-3)=-2y\Rightarrow x=\frac{-2y}{y-3}$

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Example Question #9 : Simplifying Algebraic Expressions

Simplify

(2+7x)(2-7x)

Possible Answers:

4-7x^2

4+7x^2

4+49x^2

4-49x^2

Correct answer:

4-49x^2

Explanation:

Foil

(2+7x)(2-7x)

4+14x-14x-49x^2

4-49x^2

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Example Question #1 : Simplifying Algebraic Expressions

Which answer is equivalent to $x^{40}$ ?

Possible Answers:

$(x^{20}+x^{20})$

$(x^4)^{10}$

$(x^{20})^{20}$

$(x^{20})*2$

Correct answer:

$(x^4)^{10}$

Explanation:

$(x^a)^b=x^{ab}$

Therefore:

$(x^4)^{10}=x^{40}$

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GMAT Math : Simplifying Algebraic Expressions

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Example Question #1 : Simplifying Algebraic Expressions

Example Question #1 : Simplifying Algebraic Expressions

Example Question #1 : Simplifying Algebraic Expressions

Example Question #4 : Simplifying Algebraic Expressions

Example Question #1 : Simplifying Algebraic Expressions

Example Question #6 : Simplifying Algebraic Expressions

Example Question #7 : Simplifying Algebraic Expressions

Example Question #231 : Algebra

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